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\centerline{Mathematics 132 -- Calculus for Physical and Life Sciences 2}
\centerline{Exam 1 -- Review Sheet}
\centerline{February 13, 2008}
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\noindent
{\it General Information}
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As announced in the course syllabus, the first 
midterm exam of the semester will be given (``main seating'') 
at 6:00 p.m. on Wednesday, February 20.  
The format will be similar to that of the midterm exams
last semester.  
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\item{$\bullet$}  {\bf Bring a photo ID to the exam.}
\item{$\bullet$}  The exam will be designed to take an hour but you will have an extra 
30 minutes to work and check your solutions.
\item{$\bullet$}  You will be given a TI-30 scientific calculator for the exam which 
does NOT have graphing capabilities so be prepared to answer questions without 
your personal calculator.  (Note: Some of you may have one of these calculators
purchased for use in Chemistry courses here.  That is also OK.)
\item{$\bullet$} Use of cell phones, I-pods, and all other electronic devices
{\it is not allowed} during the exam.  Please leave such devices in your room 
or put them away in your backpack (make sure cell phones are turned off).
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\noindent
{\it What will be covered}
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The exam will cover the material since the start
of the semester, including the following material
from sections 5.1 through 5.8 of Stewart:
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\item{1)}  Riemann sums and the definition of the definite integral. 
\item{2)}  The Fundamental Theorem of Calculus:  Part 1:
If $f(t)$ is continuous on $[a,b]$ and $F(x) = \int_a^x f(t)\ dt$,
then $F'(x) = f(x)$.   Part 2: (the Evaluation Theorem) If $F(x)$
is an antiderivative of a continuous function $f(x)$ on $[a,b]$, then 
$$\int_a^b f(x)\ dx = F(b) - F(a)$$
\item{3)}  Antiderivatives graphically and numerically
\item{4)}  Basic antiderivative rules: All rules coming from
basic derivative formulas:  Know $\int x^n\ dx$,
$\int a^x\ dx$, $\int \sin(x)\ dx$, $\int \cos(x)\ dx$,
$\int {1\over x^2 + 1}\ dx$, $\int {1\over \sqrt{1 - x^2}}\ dx$,
and so forth, plus the
sum, and constant multiple rules 
\item{5)}  Integrals by substitution
\item{6)}  Integrals by parts: Selecting appropriate
$u$ and $dv$, computing $du$ and $v$, using the parts formula
$\int u\ dv = uv - \int v\ du$, then finishing the integral on 
the right.  Recall that this might involve using parts again,
or another method such as substitution.
\item{7)} Integration by trigonometric substitution.
\item{8)} Integration of rational functions by partial 
fractions (see summary distributed in class).
\item{9)}  Integrals by the table:  Recognizing the appropriate
entry, using reduction formulas repeatedly if necessary.
Be aware that a preliminary substitution might be necessary
to take an integral to one of the forms in the table.
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\noindent
Note:  Some problems may ask you to carry out a particular
integration method on a problem.  Others may leave the choice
up to you.  {\it Be prepared for both types of questions!}
There will be a review for the exam in class on Tuesday, February 19.
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\noindent
{\it Review Problems}
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The Review Problems 1 - 46 at the end of Chapter 5 are good
for preparation for this exam.  It's not necessary to work
all of them.  But you should try a good selection and practice choosing 
a method at least for most of the integrals in problems 9 - 46.
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\noindent
{\it Sample Exam Questions}
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\noindent
I.  Let 
$$f(x) = \cases{1 & if $0 \le x \le 3$\cr
                x - 2 & if $3\le x \le 5$\cr
                13 - 2x & if $5 \le x \le 8$\cr}$$
\item{A)}  Sketch the graph $y = f(x)$.
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In the rest of the parts, $F(x) = \int_0^x f(t)\ dt$, where $f$ is the 
function from part A.  
\item{B)}  Assuming $F(0) = 0$, Compute $F(1), F(2), F(3), F(4), F(5), F(6), F(7), F(8)$ given 
the information in the graph of $f$.
\item{C)}  Are there any critical points of $F$?  If so, find them and say whether 
they are local maxima, local minima, or neither.  If not, say why not.
\item{D)}  Sketch the graph $y = F(x)$ if $F(0) = 0$,
and also if $F(0) = 2$.
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\noindent
II.  Find the derivatives of the following functions
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\item{A)}  $f(x) = \int_0^x \sin(t)/t\, dt$.
\item{B)}  $g(x) = \int_5^{x^3} \tan^4(t)\ dt$.
\item{C)}  $h(x) = \int_{-3x}^{5x} e^{t^2} \sin(t)\ dt$. 
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III. 
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\item{A)}  Compute $\int 5x^4 - 3\sqrt{x} + e^x + {2\over x}\ dx$
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\item{B)}  Apply a $u$-substitution to compute $\int x(4x^2 - 3)^{3/5}\ dx$
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\item{C)} Apply a $u$-substitution to compute $\int_1^2 e^{\sin(\pi x)}\cos(\pi x)\ dx$
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\item{D)}  Do you need partial fractions to 
compute $$\int {t^2 + 1\over t^3 + 3t + 3}\ dt?$$
Explain, and give a simpler method.
\item{E)} Apply integration by parts to compute $\int x^2 e^{-2x}\ dx$
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\item{F)}  Apply partial fraction decomposition to 
compute $$\int {1\over x(x - 1)(x + 2)}\ dx$$
\item{G)}  Which trigonometric substitution would you apply
to compute $\int {1\over u \sqrt{a^2 - u^2}}\ du$?
What trigonometric integral do you get after making the substitution?
Complete the derivation of the integral.
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IV.  Compute each of the integrals below using some combination of
basic rules, substitution, integration by parts, the table of integrals, 
partial fractions, and trigonometric substition. You must show all work
for full credit.  
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\item{A)}  $$\int {x^3 \over x^2 + 4x}\ dx$$
\item{B)}  $$\int (x^2 + 2x)^{3/2}\ dx$$
\item{C)}  $$\int {e^{\sqrt{\sin(x)}} \cos(x) \over \sqrt{\sin(x)}}\ dx$$
\item{D)}  $$\int {dz \over z^2\sqrt{z^2 + 6z + 10}}$$
\bye
